A generalization of Study’s theorem on convex maps

Abstract

Study proved that disks of radius r, 0 > r ⩽ 1 0 > r \leqslant 1 , get mapped onto convex (starlike) sets under all convex (starlike) maps of the unit disk U. Pommerenke and Heins gave a geometric characterization of the subsets of U that get mapped onto convex sets under all convex maps. In this note we give an analytical characterization of the subsets of U that get mapped onto α \alpha -starlike domains under all α \alpha -starlike maps of U for all α ⩾ 0 \alpha \geqslant 0 noting that 0-starlikeness equals starlikeness and 1-starlikeness equals convexity.